1 SUBROUTINE STPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
2 $ XACT, LDXACT, FERR, BERR, RESLTS )
3 *
4 * -- LAPACK test routine (version 3.1) --
5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
6 * November 2006
7 *
8 * .. Scalar Arguments ..
9 CHARACTER DIAG, TRANS, UPLO
10 INTEGER LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 REAL AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
14 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * STPT05 tests the error bounds from iterative refinement for the
21 * computed solution to a system of equations A*X = B, where A is a
22 * triangular matrix in packed storage format.
23 *
24 * RESLTS(1) = test of the error bound
25 * = norm(X - XACT) / ( norm(X) * FERR )
26 *
27 * A large value is returned if this ratio is not less than one.
28 *
29 * RESLTS(2) = residual from the iterative refinement routine
30 * = the maximum of BERR / ( (n+1)*EPS + (*) ), where
31 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
32 *
33 * Arguments
34 * =========
35 *
36 * UPLO (input) CHARACTER*1
37 * Specifies whether the matrix A is upper or lower triangular.
38 * = 'U': Upper triangular
39 * = 'L': Lower triangular
40 *
41 * TRANS (input) CHARACTER*1
42 * Specifies the form of the system of equations.
43 * = 'N': A * X = B (No transpose)
44 * = 'T': A'* X = B (Transpose)
45 * = 'C': A'* X = B (Conjugate transpose = Transpose)
46 *
47 * DIAG (input) CHARACTER*1
48 * Specifies whether or not the matrix A is unit triangular.
49 * = 'N': Non-unit triangular
50 * = 'U': Unit triangular
51 *
52 * N (input) INTEGER
53 * The number of rows of the matrices X, B, and XACT, and the
54 * order of the matrix A. N >= 0.
55 *
56 * NRHS (input) INTEGER
57 * The number of columns of the matrices X, B, and XACT.
58 * NRHS >= 0.
59 *
60 * AP (input) REAL array, dimension (N*(N+1)/2)
61 * The upper or lower triangular matrix A, packed columnwise in
62 * a linear array. The j-th column of A is stored in the array
63 * AP as follows:
64 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
65 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
66 * If DIAG = 'U', the diagonal elements of A are not referenced
67 * and are assumed to be 1.
68 *
69 * B (input) REAL array, dimension (LDB,NRHS)
70 * The right hand side vectors for the system of linear
71 * equations.
72 *
73 * LDB (input) INTEGER
74 * The leading dimension of the array B. LDB >= max(1,N).
75 *
76 * X (input) REAL array, dimension (LDX,NRHS)
77 * The computed solution vectors. Each vector is stored as a
78 * column of the matrix X.
79 *
80 * LDX (input) INTEGER
81 * The leading dimension of the array X. LDX >= max(1,N).
82 *
83 * XACT (input) REAL array, dimension (LDX,NRHS)
84 * The exact solution vectors. Each vector is stored as a
85 * column of the matrix XACT.
86 *
87 * LDXACT (input) INTEGER
88 * The leading dimension of the array XACT. LDXACT >= max(1,N).
89 *
90 * FERR (input) REAL array, dimension (NRHS)
91 * The estimated forward error bounds for each solution vector
92 * X. If XTRUE is the true solution, FERR bounds the magnitude
93 * of the largest entry in (X - XTRUE) divided by the magnitude
94 * of the largest entry in X.
95 *
96 * BERR (input) REAL array, dimension (NRHS)
97 * The componentwise relative backward error of each solution
98 * vector (i.e., the smallest relative change in any entry of A
99 * or B that makes X an exact solution).
100 *
101 * RESLTS (output) REAL array, dimension (2)
102 * The maximum over the NRHS solution vectors of the ratios:
103 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
104 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
105 *
106 * =====================================================================
107 *
108 * .. Parameters ..
109 REAL ZERO, ONE
110 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
111 * ..
112 * .. Local Scalars ..
113 LOGICAL NOTRAN, UNIT, UPPER
114 INTEGER I, IFU, IMAX, J, JC, K
115 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
116 * ..
117 * .. External Functions ..
118 LOGICAL LSAME
119 INTEGER ISAMAX
120 REAL SLAMCH
121 EXTERNAL LSAME, ISAMAX, SLAMCH
122 * ..
123 * .. Intrinsic Functions ..
124 INTRINSIC ABS, MAX, MIN
125 * ..
126 * .. Executable Statements ..
127 *
128 * Quick exit if N = 0 or NRHS = 0.
129 *
130 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
131 RESLTS( 1 ) = ZERO
132 RESLTS( 2 ) = ZERO
133 RETURN
134 END IF
135 *
136 EPS = SLAMCH( 'Epsilon' )
137 UNFL = SLAMCH( 'Safe minimum' )
138 OVFL = ONE / UNFL
139 UPPER = LSAME( UPLO, 'U' )
140 NOTRAN = LSAME( TRANS, 'N' )
141 UNIT = LSAME( DIAG, 'U' )
142 *
143 * Test 1: Compute the maximum of
144 * norm(X - XACT) / ( norm(X) * FERR )
145 * over all the vectors X and XACT using the infinity-norm.
146 *
147 ERRBND = ZERO
148 DO 30 J = 1, NRHS
149 IMAX = ISAMAX( N, X( 1, J ), 1 )
150 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
151 DIFF = ZERO
152 DO 10 I = 1, N
153 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
154 10 CONTINUE
155 *
156 IF( XNORM.GT.ONE ) THEN
157 GO TO 20
158 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
159 GO TO 20
160 ELSE
161 ERRBND = ONE / EPS
162 GO TO 30
163 END IF
164 *
165 20 CONTINUE
166 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
167 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
168 ELSE
169 ERRBND = ONE / EPS
170 END IF
171 30 CONTINUE
172 RESLTS( 1 ) = ERRBND
173 *
174 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
175 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
176 *
177 IFU = 0
178 IF( UNIT )
179 $ IFU = 1
180 DO 90 K = 1, NRHS
181 DO 80 I = 1, N
182 TMP = ABS( B( I, K ) )
183 IF( UPPER ) THEN
184 JC = ( ( I-1 )*I ) / 2
185 IF( .NOT.NOTRAN ) THEN
186 DO 40 J = 1, I - IFU
187 TMP = TMP + ABS( AP( JC+J ) )*ABS( X( J, K ) )
188 40 CONTINUE
189 IF( UNIT )
190 $ TMP = TMP + ABS( X( I, K ) )
191 ELSE
192 JC = JC + I
193 IF( UNIT ) THEN
194 TMP = TMP + ABS( X( I, K ) )
195 JC = JC + I
196 END IF
197 DO 50 J = I + IFU, N
198 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) )
199 JC = JC + J
200 50 CONTINUE
201 END IF
202 ELSE
203 IF( NOTRAN ) THEN
204 JC = I
205 DO 60 J = 1, I - IFU
206 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) )
207 JC = JC + N - J
208 60 CONTINUE
209 IF( UNIT )
210 $ TMP = TMP + ABS( X( I, K ) )
211 ELSE
212 JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2
213 IF( UNIT )
214 $ TMP = TMP + ABS( X( I, K ) )
215 DO 70 J = I + IFU, N
216 TMP = TMP + ABS( AP( JC+J-I ) )*ABS( X( J, K ) )
217 70 CONTINUE
218 END IF
219 END IF
220 IF( I.EQ.1 ) THEN
221 AXBI = TMP
222 ELSE
223 AXBI = MIN( AXBI, TMP )
224 END IF
225 80 CONTINUE
226 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
227 $ MAX( AXBI, ( N+1 )*UNFL ) )
228 IF( K.EQ.1 ) THEN
229 RESLTS( 2 ) = TMP
230 ELSE
231 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
232 END IF
233 90 CONTINUE
234 *
235 RETURN
236 *
237 * End of STPT05
238 *
239 END
2 $ XACT, LDXACT, FERR, BERR, RESLTS )
3 *
4 * -- LAPACK test routine (version 3.1) --
5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
6 * November 2006
7 *
8 * .. Scalar Arguments ..
9 CHARACTER DIAG, TRANS, UPLO
10 INTEGER LDB, LDX, LDXACT, N, NRHS
11 * ..
12 * .. Array Arguments ..
13 REAL AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
14 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
15 * ..
16 *
17 * Purpose
18 * =======
19 *
20 * STPT05 tests the error bounds from iterative refinement for the
21 * computed solution to a system of equations A*X = B, where A is a
22 * triangular matrix in packed storage format.
23 *
24 * RESLTS(1) = test of the error bound
25 * = norm(X - XACT) / ( norm(X) * FERR )
26 *
27 * A large value is returned if this ratio is not less than one.
28 *
29 * RESLTS(2) = residual from the iterative refinement routine
30 * = the maximum of BERR / ( (n+1)*EPS + (*) ), where
31 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
32 *
33 * Arguments
34 * =========
35 *
36 * UPLO (input) CHARACTER*1
37 * Specifies whether the matrix A is upper or lower triangular.
38 * = 'U': Upper triangular
39 * = 'L': Lower triangular
40 *
41 * TRANS (input) CHARACTER*1
42 * Specifies the form of the system of equations.
43 * = 'N': A * X = B (No transpose)
44 * = 'T': A'* X = B (Transpose)
45 * = 'C': A'* X = B (Conjugate transpose = Transpose)
46 *
47 * DIAG (input) CHARACTER*1
48 * Specifies whether or not the matrix A is unit triangular.
49 * = 'N': Non-unit triangular
50 * = 'U': Unit triangular
51 *
52 * N (input) INTEGER
53 * The number of rows of the matrices X, B, and XACT, and the
54 * order of the matrix A. N >= 0.
55 *
56 * NRHS (input) INTEGER
57 * The number of columns of the matrices X, B, and XACT.
58 * NRHS >= 0.
59 *
60 * AP (input) REAL array, dimension (N*(N+1)/2)
61 * The upper or lower triangular matrix A, packed columnwise in
62 * a linear array. The j-th column of A is stored in the array
63 * AP as follows:
64 * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
65 * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
66 * If DIAG = 'U', the diagonal elements of A are not referenced
67 * and are assumed to be 1.
68 *
69 * B (input) REAL array, dimension (LDB,NRHS)
70 * The right hand side vectors for the system of linear
71 * equations.
72 *
73 * LDB (input) INTEGER
74 * The leading dimension of the array B. LDB >= max(1,N).
75 *
76 * X (input) REAL array, dimension (LDX,NRHS)
77 * The computed solution vectors. Each vector is stored as a
78 * column of the matrix X.
79 *
80 * LDX (input) INTEGER
81 * The leading dimension of the array X. LDX >= max(1,N).
82 *
83 * XACT (input) REAL array, dimension (LDX,NRHS)
84 * The exact solution vectors. Each vector is stored as a
85 * column of the matrix XACT.
86 *
87 * LDXACT (input) INTEGER
88 * The leading dimension of the array XACT. LDXACT >= max(1,N).
89 *
90 * FERR (input) REAL array, dimension (NRHS)
91 * The estimated forward error bounds for each solution vector
92 * X. If XTRUE is the true solution, FERR bounds the magnitude
93 * of the largest entry in (X - XTRUE) divided by the magnitude
94 * of the largest entry in X.
95 *
96 * BERR (input) REAL array, dimension (NRHS)
97 * The componentwise relative backward error of each solution
98 * vector (i.e., the smallest relative change in any entry of A
99 * or B that makes X an exact solution).
100 *
101 * RESLTS (output) REAL array, dimension (2)
102 * The maximum over the NRHS solution vectors of the ratios:
103 * RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
104 * RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
105 *
106 * =====================================================================
107 *
108 * .. Parameters ..
109 REAL ZERO, ONE
110 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
111 * ..
112 * .. Local Scalars ..
113 LOGICAL NOTRAN, UNIT, UPPER
114 INTEGER I, IFU, IMAX, J, JC, K
115 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
116 * ..
117 * .. External Functions ..
118 LOGICAL LSAME
119 INTEGER ISAMAX
120 REAL SLAMCH
121 EXTERNAL LSAME, ISAMAX, SLAMCH
122 * ..
123 * .. Intrinsic Functions ..
124 INTRINSIC ABS, MAX, MIN
125 * ..
126 * .. Executable Statements ..
127 *
128 * Quick exit if N = 0 or NRHS = 0.
129 *
130 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
131 RESLTS( 1 ) = ZERO
132 RESLTS( 2 ) = ZERO
133 RETURN
134 END IF
135 *
136 EPS = SLAMCH( 'Epsilon' )
137 UNFL = SLAMCH( 'Safe minimum' )
138 OVFL = ONE / UNFL
139 UPPER = LSAME( UPLO, 'U' )
140 NOTRAN = LSAME( TRANS, 'N' )
141 UNIT = LSAME( DIAG, 'U' )
142 *
143 * Test 1: Compute the maximum of
144 * norm(X - XACT) / ( norm(X) * FERR )
145 * over all the vectors X and XACT using the infinity-norm.
146 *
147 ERRBND = ZERO
148 DO 30 J = 1, NRHS
149 IMAX = ISAMAX( N, X( 1, J ), 1 )
150 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
151 DIFF = ZERO
152 DO 10 I = 1, N
153 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
154 10 CONTINUE
155 *
156 IF( XNORM.GT.ONE ) THEN
157 GO TO 20
158 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
159 GO TO 20
160 ELSE
161 ERRBND = ONE / EPS
162 GO TO 30
163 END IF
164 *
165 20 CONTINUE
166 IF( DIFF / XNORM.LE.FERR( J ) ) THEN
167 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
168 ELSE
169 ERRBND = ONE / EPS
170 END IF
171 30 CONTINUE
172 RESLTS( 1 ) = ERRBND
173 *
174 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
175 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
176 *
177 IFU = 0
178 IF( UNIT )
179 $ IFU = 1
180 DO 90 K = 1, NRHS
181 DO 80 I = 1, N
182 TMP = ABS( B( I, K ) )
183 IF( UPPER ) THEN
184 JC = ( ( I-1 )*I ) / 2
185 IF( .NOT.NOTRAN ) THEN
186 DO 40 J = 1, I - IFU
187 TMP = TMP + ABS( AP( JC+J ) )*ABS( X( J, K ) )
188 40 CONTINUE
189 IF( UNIT )
190 $ TMP = TMP + ABS( X( I, K ) )
191 ELSE
192 JC = JC + I
193 IF( UNIT ) THEN
194 TMP = TMP + ABS( X( I, K ) )
195 JC = JC + I
196 END IF
197 DO 50 J = I + IFU, N
198 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) )
199 JC = JC + J
200 50 CONTINUE
201 END IF
202 ELSE
203 IF( NOTRAN ) THEN
204 JC = I
205 DO 60 J = 1, I - IFU
206 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) )
207 JC = JC + N - J
208 60 CONTINUE
209 IF( UNIT )
210 $ TMP = TMP + ABS( X( I, K ) )
211 ELSE
212 JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2
213 IF( UNIT )
214 $ TMP = TMP + ABS( X( I, K ) )
215 DO 70 J = I + IFU, N
216 TMP = TMP + ABS( AP( JC+J-I ) )*ABS( X( J, K ) )
217 70 CONTINUE
218 END IF
219 END IF
220 IF( I.EQ.1 ) THEN
221 AXBI = TMP
222 ELSE
223 AXBI = MIN( AXBI, TMP )
224 END IF
225 80 CONTINUE
226 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
227 $ MAX( AXBI, ( N+1 )*UNFL ) )
228 IF( K.EQ.1 ) THEN
229 RESLTS( 2 ) = TMP
230 ELSE
231 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
232 END IF
233 90 CONTINUE
234 *
235 RETURN
236 *
237 * End of STPT05
238 *
239 END